The chirp modulation method is a modulation method in which the frequency of a signal (chirp) varies linearly over time in a bandwidth of Fs Hz. A chirp having a positive gradient in the frequency-time plane is generally referred to as an up-chirp, for example chirp 1 and chirp 2 on FIG. 1. A chirp having a negative gradient in the frequency-time plane is generally referred to as a down-chirp, for example chirp 3 on FIG. 1.
A chirp can be represented by a sequence of N samples. One or more identical contiguous chirps can form a symbol that represents a data value to be communicated. A chirp can be represented mathematically as:C(g,p)=ejπg(n−fn(p))(n+1−fn(p))/N  (equation 1)where g is the gradient of the chirp, N is the number of samples in the sequence, n is a sample in the sequence, p is the symbol's value, fn(p) is a function that encodes p onto the received chirp, which implicitly may also be a function of g, n, N and other constants, and C is the received chirp sequence, which is normally evaluated for all integer values of n from 0 to N−1 in order. The number of valid values of p is the symbol set size, which is nominally N. However, the symbol set size can be more or less than N depending on the quality of the link. The value of g can have any value greater than 0 and less than N. Preferably, g is an integer between 1 and N−1. Due to the modular nature of this expression negative gradients are obtained from N−1 backwards. Hence, N−2 is equivalent to a negative gradient of −2. Where there are more than one identical contiguous chirps in a symbol, each chirp individually conveys the same value which is the symbol value of the symbol.
Chirp 1 in FIG. 1 has a starting frequency of −Fs/2 and a gradient of 1. It increases linearly in frequency over a period of N samples at a sample rate of Fs to reach a frequency close to +Fs/2. Since this is a complex sampled system +Fs/2 is the same as −Fs/2. Multiple chirps are usually contiguous but may start with a different frequency. The signal phase is typically made continuous throughout a sequence of chirps. In other words, after the signal has reached +Fs/2 at n=N−1, the next symbol starts with n=0 again. FIG. 1 illustrates an example in which two consecutive chirps have the same symbol value, whereas the third chirp is different. An apparent discontinuity in frequency between chirp 1 and chirp 2 occurs at n=N.
Chirp 4 in FIG. 2 has a gradient of 2 and a starting frequency of −Fs/2. Because it has double the gradient of the chirps of FIG. 1, it increases linearly in frequency to +Fs/2 in half the number of samples that the chirps in FIG. 1 do, i.e. it reaches close to +Fs/2 after close to N/2 samples. The chirp then wraps around in frequency. Since this is a sampled system, these frequency wraps are in effect continuous and have continuous phase. The chirp repeats the frequency sweep from −Fs/2 to +Fs/2 between samples N/2 and N.
The chirps also have continuous frequency and phase from one end of the chirp to the other. A cyclic shift of the samples that make up a chirp creates another valid chirp.
In typical communication systems where privacy is required on a link between devices, communications on that link are encrypted. Generally this involves establishing a basic public connection, and then exchanging security keys (for example using the D-H key exchange algorithm). An XOR operation is performed with the data to be communicated and the security key to generate the encrypted signal. On receipt of this encrypted signal, the receiver performs an XOR operation with the encrypted signal and the security key in order to decrypt the data. Typically, security keys are exchanged for every message to be transmitted privately. Typically, the encrypted message bits that are transmitted can easily be received by a third party. However, in the absence of the key the third party will find it hard to decrypt the message.
This encryption method is suitable for systems utilising high data rates, and operating on devices having large energy reserves. However, chirp communications are typically used in systems operating using low data rates and short messages. The exchange of security keys for every message to be sent would cause significant delay when operating using low data rate chirp signals. Additionally, chirps signals are typically communicated between low power devices, for example battery powered handheld devices. The processing power required to communicate using the encryption process described above would be a significant drain of power for a low power device.
In alternative encrypted communication systems that use low power and short messages, the key exchange may be performed infrequently or only once, but otherwise operates as previously described. Typically, this is achieved by bringing the two devices into close proximity so that the keys can be exchanged securely. Alternatively, the D-H algorithm is used as previously described, but used less often. In particular, with a broadcast system, the keys cannot be modified after every message as there is no feedback to guarantee that both transmitter and receiver have the same key. Typically, the key must remain fixed or is a function of the public part of a message. A disadvantage of this approach is that a third party listening to the communication would still be able to read the bits of the communication and over time guess the security key, either directly or via the public part of a message. For example, if one knew that the transmitted data were speech, then with enough time, one could try decoding a message with all possible keys until something resembling speech resulted. This process is made easier if all messages share the same key. This process can be automated and has been shown to break some encryption methods.
Thus, there is a need for an improved method of encrypting chirp communications which is less power intensive than traditional encryption methods, and where the encrypted data bits themselves cannot easily be read, and is suitable for systems operating using low data rates and short messages.